A369759 - OEIS

%I #8 Feb 02 2024 11:01:51

%S 1,9,28,9,126,252,344,9,28,1134,1332,252,2198,3096,3528,33,4914,252,

%T 6860,1134,9632,11988,12168,252,126,19782,28,3096,24390,31752,29792,

%U 33,37296,44226,43344,252,50654,61740,61544,1134,68922,86688,79508,11988,3528,109512

%N The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.

%H Amiram Eldar, <a href="/A369759/b369759.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A356192(n)).

%F Multiplicative with a(p) = p^3 + 1, a(p^e) = p^e + 1 for an odd e >= 3, and a(p^e) = p^(e+1) + 1 for an even e.

%F a(n) >= A034448(n), with equality if and only if n is cubefull exponentially odd number (A335988).

%F Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(2*s-2) - 1/p^(3*s-5) + 1/p^(4*s-5) - 1/p^(4*s-3)).

%F Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (zeta(4)*zeta(6)/zeta(2)) * Product_{p prime} (1 - 1/p^6 + 1/p^11 - 1/p^12) = 0.65813930591740259189... .

%t f[p_, e_] := p^If[OddQ[e], Max[e, 3], e+1] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i,1]^if(f[i,2]%2, max(f[i,2], 3), f[i,2] + 1));}

%Y Cf. A034448, A335988, A356192, A365480, A369757, A369758.

%Y Cf. A013661, A013662, A013664.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 31 2024